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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfxnegmnf2 | Structured version Visualization version GIF version |
Description: A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2.j | ⊢ Ⅎ𝑗𝐹 |
xlimpnfxnegmnf2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimpnfxnegmnf2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimpnfxnegmnf2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2 | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimpnfxnegmnf2.j | . . 3 ⊢ Ⅎ𝑗𝐹 | |
2 | xlimpnfxnegmnf2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimpnfxnegmnf2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | xlimpnfxnegmnf 42456 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
5 | xlimpnfxnegmnf2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 1, 5, 2, 3 | xlimpnf 42484 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
7 | nfmpt1 5128 | . . . 4 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) | |
8 | 3 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*) |
9 | 8 | xnegcld 12681 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝑒(𝐹‘𝑘) ∈ ℝ*) |
10 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑘-𝑒(𝐹‘𝑗) | |
11 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑗𝑘 | |
12 | 1, 11 | nffv 6655 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐹‘𝑘) |
13 | 12 | nfxneg 42100 | . . . . . 6 ⊢ Ⅎ𝑗-𝑒(𝐹‘𝑘) |
14 | fveq2 6645 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
15 | 14 | xnegeqd 42074 | . . . . . 6 ⊢ (𝑗 = 𝑘 → -𝑒(𝐹‘𝑗) = -𝑒(𝐹‘𝑘)) |
16 | 10, 13, 15 | cbvmpt 5131 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)) |
17 | 9, 16 | fmptd 6855 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)):𝑍⟶ℝ*) |
18 | 7, 5, 2, 17 | xlimmnf 42483 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥)) |
19 | 2 | uztrn2 12250 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍) |
20 | xnegex 12589 | . . . . . . . . 9 ⊢ -𝑒(𝐹‘𝑗) ∈ V | |
21 | fvmpt4 41874 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ -𝑒(𝐹‘𝑗) ∈ V) → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) | |
22 | 20, 21 | mpan2 690 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) |
23 | 22 | breq1d 5040 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
24 | 19, 23 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
25 | 24 | ralbidva 3161 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → (∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
26 | 25 | rexbiia 3209 | . . . 4 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
27 | 26 | ralbii 3133 | . . 3 ⊢ (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
28 | 18, 27 | syl6bb 290 | . 2 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
29 | 4, 6, 28 | 3bitr4d 314 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 -𝑒cxne 12492 ~~>*clsxlim 42460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-z 11970 df-uz 12232 df-xneg 12495 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-topgen 16709 df-ordt 16766 df-ps 17802 df-tsr 17803 df-top 21499 df-topon 21516 df-bases 21551 df-lm 21834 df-xlim 42461 |
This theorem is referenced by: (None) |
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